It's always the frustration of Shakespeare that, although his words are everywhere, the man is invisible. Anonymous exploits this, inviting audiences to entertain the ultimate plot: that the writer of Macbeth or The Tempest is not, in fact, a man named William Shakespeare, but …

Well, why not? All we know for certain is that Shaxpere, Shaxberd, or Shakspear (he spelled his name in as many as 25 different ways) was born in Stratford-upon-Avon in 1564, that he was an actor whose name was printed, with the names of his fellow performers, in the collected edition of his plays in 1623. We know that he married Anne Hathaway and died in 1616, according to legend on his birthday, which happened to be St George's Day. What's known as the "Stratfordian" case for Shakespeare rests on these and a handful of other facts, but, basically, that's it.

Yet this shadowy author's work has become a global phenomenon. According to a recent issue of the New York Times, "Shakespeare surrounds us this season, perhaps even more than usual. On any given summer's day, somewhere in America, you can find fairies frolicking alongside young lovers in flight from Athens in A Midsummer Night's Dream."

Forget the frolicking, watch the potential box office. In 1998, Shakespeare in Love, Tom Stoppard's witty and ingenious historical romcom about the poet's rivalry with Christopher Marlowe, starring Joseph Fiennes and Gwyneth Paltrow, scooped seven Academy Awards and grossed more than £60m. Globally, Shakespeare plc has an annual turnover of several billion dollars. He's one of our biggest exports.

The identity of the writer who has such a sure grip of his audience has puzzled commentators for centuries. Even in his own day, long before he had emerged as "William Shakespeare", the playwright drove people mad with his modest Stratford origins. In 1592, rival dramatist Robert Greene made a celebrated deathbed attack on the "conceit" of the "upstart crow" from the provinces who, with intolerable airs and graces, considered himself "the only Shake-scene". For Greene and every subsequent Shakespeare naysayer, there is something enraging about the poet's genius. The explanation must be that Shakespeare is not – simply cannot be – original, but an impostor "beautified with our feathers". Snootily, how could someone born and raised in provincial Stratford have such a command of language, plots and ideas ?

Over four centuries, the mystery of William Shakespeare (whose singular genius some compare to Mozart's) has animated several increasingly bizarre conspiracy theories. There was such an unbridgeable chasm between the complex brilliance of the plays and what they suggest about their author's education and experience, on the one hand, set against the bare facts of Shakespeare's life, on the other, that a better explanation of his "genius" had to be found.

It was impossible, said the "anti-Stratfordians", as the sceptics came to be known, that the recorded life of the man called Shakespeare could yield the astonishing universality and dazzling inventiveness of the canon. Anonymous is the latest, quixotic attempt to fill this vacuum and to create a "Shakespeare" that's the work of a more obviously accomplished writer, a certain Edward de Vere.

The Earl of Oxford, as he was, is first among a roll call that includes Sir Francis Bacon, Christopher Marlowe, Sir Walter Raleigh, John Donne and even Elizabeth I, the virgin queen herself, who makes a walk-on appearance in Anonymous, played by Vanessa Redgrave. Actually, Emmerich is in good company. The unholy alliance of "anti-Stratfordians" boasts Mark Twain, Charlie Chaplin, Orson Welles and Sigmund Freud among its members. In the British theatre today, Mark Rylance and Derek Jacobi (who appear in Anonymous) are vociferous spokesmen for the Earl of Oxford.

Conspiracy theories are always touched with a bit of madness. Those who are convinced Edward de Vere is the real author of the Shakespeare canon – the plays, they contend, are his surrogate autobiography – have to brush aside some inconvenient truths. The most challenging of these is that the crafty earl died inconveniently young in 1604, well before Macbeth, King Lear, Coriolanus, The Winter's Tale and The Tempest were written and/or staged.

Does it matter? The world of "Shakespeare", who – or whatever – he might be, continues to revolve on stage and screen and across the traffic of countless web pages. Audiences, actors and directors are still drawn to "Shakespeare plays", whose stories flow through the world's imagination from day to day. Many actors are defined in the public mind by their interpretation of Shakespeare's characters. He is a playwright who has given them their defining roles: Vanessa Redgrave's Rosalind, Chiwetel Ejiofor's Othello; Simon Russell Beale's Iago; Mark Rylance's Hamlet; Derek Jacobi's Lear. These are performances (everyone has their favourite) that find an extraordinary depth and subtlety in the playwright's "fire-new words".

Against Emmerich, these plays – the canon – have an internal consistency, a natural authenticity that makes its own "Stratfordian" argument. There are three unmistakable hallmarks to Shakespeare's writing.

First, there's his humanity, his benign capacity to find in the darkest villainy something with which the common man or woman in the pit can identify, from Macbeth's obsession with witchcraft to Richard III's antic humour.

Second, there's Shakespeare's instinctive theatricality, so different (actors always say) from the majestic speech-making of Christopher Marlowe. Arguably, only Shakespeare could have written act three of King Lear, in which an old king on the edge of insanity, his Fool and another character clad only in a blanket conduct an imaginary trial of the king's termagant daughters in a scene that is more Beckett than Bacon.

And finally, with Shakespeare, audiences are never far from a fundamental domesticity: sly allusions to his father's glove-making business; obscure Warwickshire dialect words; Ophelia picking wild flowers from the river bank; Touchstone rhapsodising about his sheep; Othello driven to jealous rage by a simple cambric handkerchief; the warring citizens of Verona in Romeo and Juliet sounding for all the world like Stratford townspeople.

All of this – what Sir Peter Hall has called "the sheer bloody Englishness of the whole thing" – makes it hard to believe that the plays were not written by one man. Who that man is, or was, will always remain a matter for debate. You can call him Anonymous, or you can call him Shakespeare. Does it really matter? He is for Everyman, and the work remains.